Estimating the growth in Mordell–Weil ranks and Shafarevich–Tate groups over Lie extensions

Delbourgo, Daniel; Lei, Antonio

doi:10.1007/s11139-016-9785-1

Cited by 12 publications

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References 28 publications

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“…This growth property for the λ-invariant is ubiquitous in non-commutative Iwasawa theory, and at weight two (i.e. for modular abelian varieties) can actually produce large Mordell-Weil ranks over the finite layers in the p-adic Lie extension -see [5,7] for situations where most of the λ-invariant is absorbed into the MW-rank.…”

Section: Theorem 1•5 Under the Same Conditions Asmentioning

confidence: 99%

Variation of the algebraic λ-invariant over a solvable extension

Delbourgo¹

2019

Math. Proc. Camb. Phil. Soc.

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Fix an odd prime p. Let $\mathcal{D}_n$ denote a non-abelian extension of a number field K such that $K\cap\mathbb{Q}(\mu_{p^{\infty}})=\mathbb{Q}, $ and whose Galois group has the form $ \text{Gal}\big(\mathcal{D}_n/K\big)\cong \big(\mathbb{Z}/p^{n'}\mathbb{Z}\big)^{\oplus g}\rtimes \big(\mathbb{Z}/p^n\mathbb{Z}\big)^{\times}\ $ where g > 0 and $0 \lt n'\leq n$ . Given a modular Galois representation $\overline{\rho}:G_{\mathbb{Q}}\rightarrow \text{GL}_2(\mathbb{F})$ which is p-ordinary and also p-distinguished, we shall write $\mathcal{H}(\overline{\rho})$ for the associated Hida family. Using Greenberg’s notion of Selmer atoms, we prove an exact formula for the algebraic λ-invariant \begin{equation} \lambda^{\text{alg}}_{\mathcal{D}_n}(f) \;=\; \text{the number of zeroes of } \text{char}_{\Lambda}\big(\text{Sel}_{\mathcal{D}_n^{\text{cy}}}\big(f\big)^{\wedge}\big) \end{equation} at all $f\in\mathcal{H}(\overline{\rho})$ , under the assumption $\mu^{\text{alg}}_{K(\mu_p)}(f_0)=0$ for at least one form f0. We can then easily deduce that $\lambda^{\text{alg}}_{\mathcal{D}_n}(f)$ is constant along branches of $\mathcal{H}(\overline{\rho})$ , generalising a theorem of Emerton, Pollack and Weston for $\lambda^{\text{alg}}_{\mathbb{Q}(\mu_{p})}(f)$ . For example, if $\mathcal{D}_{\infty}=\bigcup_{n\geq 1}\mathcal{D}_n$ has the structure of a p-adic Lie extension then our formulae include the cases where: either (i) $\mathcal{D}_{\infty}/K$ is a g-fold false Tate tower, or (ii) $\text{Gal}\big(\mathcal{D}_{\infty}/K(\mu_p)\big)$ has dimension ≤ 3 and is a pro-p-group.

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Section: Theorem 1•5 Under the Same Conditions Asmentioning

confidence: 99%

Variation of the algebraic λ-invariant over a solvable extension

Delbourgo¹

2019

Math. Proc. Camb. Phil. Soc.

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“…Since 𝐸 has bad reduction at only finitely many primes, we deduce that 𝑞 1 = 0 for all extensions 𝐹 (ℓ) ∞ except for finitely many choices of ℓ. Recall that 𝑄 2 consists of the primes 𝑤 ∤ 7 of 𝐹 cyc that are ramified in 𝐹 ∞ , such that 𝐸 has good reduction at 𝑤 and 𝐸 (𝐹 cyc,𝑤 ) [7] ≠ 0. Since the formal group of 𝐸 at 𝑤 is pro-ℓ, 𝐸 (𝐹 cyc,𝑤 ) [7] ≃ 𝐸 (𝑘 𝑤 ) [7], where 𝑘 𝑤 is the residue field of 𝐹 cyc,𝑤 .…”

Section: False-tate Curve Extensionsmentioning

confidence: 97%

“…Recall that 𝑄 2 consists of the primes 𝑤 ∤ 7 of 𝐹 cyc that are ramified in 𝐹 ∞ , such that 𝐸 has good reduction at 𝑤 and 𝐸 (𝐹 cyc,𝑤 ) [7] ≠ 0. Since the formal group of 𝐸 at 𝑤 is pro-ℓ, 𝐸 (𝐹 cyc,𝑤 ) [7] ≃ 𝐸 (𝑘 𝑤 ) [7], where 𝑘 𝑤 is the residue field of 𝐹 cyc,𝑤 . Since 𝑘 𝑤 is a 7-extension of F ℓ , it follows that 𝐸 (𝑘 𝑤 ) [7] ≠ 0 if and only if 𝐸 (F ℓ ) [7] ≠ 0.…”

Section: False-tate Curve Extensionsmentioning

confidence: 99%

“…Since the formal group of 𝐸 at 𝑤 is pro-ℓ, 𝐸 (𝐹 cyc,𝑤 ) [7] ≃ 𝐸 (𝑘 𝑤 ) [7], where 𝑘 𝑤 is the residue field of 𝐹 cyc,𝑤 . Since 𝑘 𝑤 is a 7-extension of F ℓ , it follows that 𝐸 (𝑘 𝑤 ) [7] ≠ 0 if and only if 𝐸 (F ℓ ) [7] ≠ 0. Thus, the prime 𝑤 lies in 𝑄 2 if the following conditions are satisfied:…”

Section: False-tate Curve Extensionsmentioning

confidence: 99%

“…For elliptic curves 𝐸 /𝐹 , asymptotic formulas for the growth of the rank of 𝐸 (𝐹 (𝑛) ) as 𝑛 → ∞ have been proven by A. Lei and F. Sprung in [19]. More Mordell-Weil ranks in noncommutative towers 3 recently, such growth questions are studied in admissible uniform pro-𝑝 extensions of number fields by D. Delbourgo and A. Lei in [7], and by P. C. Hung and M. F. Lim in [14].…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic growth of Mordell–Weil ranks of elliptic curves in noncommutative towers

Ray

2022

Can. Math. Bull.

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Let 𝐸 be an elliptic curve defined over a number field 𝐹 with good ordinary reduction at all primes above 𝑝, and let 𝐹 ∞ be a finitely ramified uniform pro-𝑝 extension of 𝐹 containing the cyclotomic Z 𝑝 -extension 𝐹 cyc . Set 𝐹 (𝑛) be the 𝑛-th layer of the tower, and 𝐹 (𝑛) cyc the cyclotomic Z 𝑝 -extension of 𝐹 (𝑛) . We study the growth of the rank of 𝐸 (𝐹 (𝑛) ) by analyzing the growth of the 𝜆-invariant of the Selmer group over 𝐹 (𝑛) cyc as 𝑛 → ∞. This method has its origins in work of A. Cuoco, who studied Z 2 𝑝 -extensions. Refined estimates for growth are proved that are close to conjectured estimates. The results are illustrated in special cases. 2022/01/21 16:40 This is a ``preproof'' accepted article for Canadian Mathematical Bulletin This version may be subject to change during the production process.

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Rational points on algebraic curves in infinite towers of number fields

Ray

2022

Ramanujan J

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Estimating the growth in Mordell–Weil ranks and Shafarevich–Tate groups over Lie extensions

Cited by 12 publications

References 28 publications

Variation of the algebraic λ-invariant over a solvable extension

Variation of the algebraic λ-invariant over a solvable extension

Asymptotic growth of Mordell–Weil ranks of elliptic curves in noncommutative towers

Rational points on algebraic curves in infinite towers of number fields

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